Magnetic Field Lines

The magnetic field surrounding a bar-magnet (for example) is continuous. The lines that are drawn to depict the strength and direction of the field don't actually exist. They are akin to drawing contours on an OS map, are they not? What therefore explains the behaviour of iron filings when they are scattered in the magnetic field of a bar magnet? Why do the arrange themselves along distinct 'lines', creating a pattern of peaks and troughs in the field? Does continuum or statistical mechanics offer an explanation the observed behaviour?

The magnetic field surrounding a bar-magnet (for example) is continuous. The lines that are drawn to depict the strength and direction of the field don't actually exist. They are akin to drawing contours on an OS map, are they not? What therefore explains the behaviour of iron filings when they are scattered in the magnetic field of a bar magnet? Why do the arrange themselves along distinct 'lines', creating a pattern of peaks and troughs in the field? Does continuum or statistical mechanics offer an explanation the observed behaviour?

Yes, field lines are like a contour map. The explanation of the nifty behavior of iron filings is that when either the filings are longish pieces of metal or alternatively small chunks stick together to form longish pieces, you find that since the field magnetizes the iron making it a tiny magnet and they all are magnetized in the direction of the field The forces on the filing tend to rotate them to align with the field direction and then the end of one (say North) will also want to stick to the other end (south) of the filing right in front of it. Hence if you add some statistical randomness (shake the paper) one gets a pattern of stuck together filings that is highly suggestive of (but is not really) field "lines". It was probably these patterns that caused early electricians (as they were called then) to invent the whole lines of force theory which really causes more confusion than clarity.